Question: A list of five positive integers has all of the following properties:

$\bullet$  The only integer in the list that occurs more than once is $8,$

$\bullet$  its median is $9,$ and

$\bullet$  its average (mean) is $10.$

What is the largest possible integer that could appear in the list?
We write the list of five numbers in increasing order. We know that the number $8$ occurs at least twice in the list. Since the median of the list is $9,$ then the middle number (that is, the third number) in the list is $9.$ Thus, the list can be written as $a,$ $b,$ $9,$ $d,$ $e.$

Since $8$ occurs more than once and the middle number is $9,$ then $8$ must occur twice only with $a=b=8.$ Thus, the list can be written as $8,$ $8,$ $9,$ $d,$ $e.$

Since the average is $10$ and there are $5$ numbers in the list, then the sum of the numbers in the list is $5(10)=50.$ Therefore, $8+8+9+d+e=50$ or $25+d+e=50$ or $d+e=25.$

Since $8$ is the only integer that occurs more than once in the list, then $d>9.$ Thus, $10 \leq d < e$ and $d+e=25.$ To make $e$ as large as possible, we make $d$ as small as possible, so we make $d=10,$ and so $e=15.$

The list $8,$ $8,$ $9,$ $10,$ $15$ has the desired properties, so the largest possible integer that could appear in the list is $\boxed{15}.$